Marek Kaluba

Assistant Professor

Biography

I am an assistant professor at the Adam Mickiewicz University from 2014.
The Mathematical Institute of Polish Academy of Sciences hosted me as a Post-Doc between 2015 and 2017, where I worked with Piotr Nowak on Kazhdan’s property (T).
My research interests include also group actions, topology of high-dimensional manifolds (surgery and equivariant surgery theory), applied topology (persistence and others) and symbolic computation.
From time to time I like to code – the effects may be (rarely) found on github or else (see links on the left).

Programming: Julia, Python
Sport: Climbing, Yoga

Interests

Property (T)

Computation in mathematics

Group actions

Topology of manifolds

Applied topology

Education

PhD in Mathematics, 2014

Adam Mickiewicz University, Poznań

MSc in Mathematics, 2010

Adam Mickiewicz University, Poznań

Selected Publications

We prove that $\operatorname{Aut}(F_n)$ has Kazhdan’s property (T) for every $n \geqslant 6$. Our proof relies on relating the Laplace operators of $\operatorname{SAut}(F_n)$ for various $n$ via symmetrisation by torsion. The method works also for $\operatorname{SL}_n(\mathbb{Z})$ and for $n \geqslant 3$ yields a new proof of the fact that $\operatorname{SL}_n(\mathbb{Z})$ has property (T).
We also provide new, explicit lower bounds for the Kazhdan constants of $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 6$) with respect to natural generating sets.

We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have Kazhdan property (T).

Given $M\times S^n$, where $M$ is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group, we try to answer how many (non-product) actions are there on such oddities?